Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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52 SHINICHI MOCHIZUKI hyperbolic orbicurve C v is of type (1, Z/lZ) ± [cf. [EtTh], Definition 2.5, (i)]. [Here, we note that it follows from the portion of (b) concerning 2-torsion points that the base field K v satisfies the assumption “K = ¨K” of [EtTh], Definition 2.5, (i).] Finally, we observe that when v ∈ V bad ,it follows from the theory of [EtTh], §2 — i.e., roughly speaking, “by taking an l-th root of the theta function” —thatX v , C v admit natural models X v , C v over K v , which are hyperbolic orbicurves of type (1, (Z/lZ) Θ ), (1, (Z/lZ) Θ ) ± , respectively [cf. [EtTh], Definition 2.5, (i)]; these models determine open subgroups Π Xv ⊆ Π Cv ⊆ Π Cv .Ifv ∈ V bad , then, relative to the notation of Remark 3.1.1 below, we shall write Π v def =Π tp X v . (f) ɛ is a cusp of the hyperbolic orbicurve C K [cf. (d)] that arises from a nonzero element of the quotient “Q” that appears in the definition of a “hyperbolic orbicurve of type (1,l-tors) ± ” given in [EtTh], Definition 2.1. If v ∈ V, then let us write ɛ v for the cusp of C v determined by ɛ. If v ∈ V bad , then we assume that ɛ v is the cusp that arises from the canonical generator [up to sign] “±1” of the quotient “Ẑ” that appears in the definition of a “hyperbolic orbicurve of type (1, Z/lZ) ± ” given in def [EtTh], Definition 2.5, (i). Thus, the data (X K = X F × F K, C K ,ɛ) determines hyperbolic orbicurves X−→ K , C −→K of type (1,l-tors −−→ ), (1,l-tors −−→ ) ±, respectively [cf. Definition 1.1, Remark 1.1.2], as well as open subgroups Π X−→K ⊆ Π C−→K ⊆ Π CF ,Δ X−→ ⊆ Δ C−→ ⊆ Δ C . If v ∈ V good def , then we shall write Π v =Π X−→v . Remark 3.1.1. Relative to the notation of Definition 3.1, (e), suppose that v ∈ V non . Then in addition to the various profinite groups Π (−)v ,Δ (−) ,onealso has corresponding tempered fundamental groups Π tp (−) v ; Δ tp (−) v [cf. [André], §4; [SemiAnbd], Example 3.10], whose profinite completions may be identified with Π (−)v ,Δ (−) . Here, we note that unlike “Δ (−) ”, the topological group Δ tp (−) v depends, a priori, on v. Remark 3.1.2. (i) Observe that the open subgroup Π XK ⊆ Π CK may be constructed grouptheoretically from the topological group Π CK . Indeed, it follows immediately from
INTER-UNIVERSAL TEICHMÜLLER THEORY I 53 the construction of the coverings “X”, “C” in the discussion at the beginning of [EtTh], §2 [cf. also [AbsAnab], Lemma 1.1.4, (i)], that the closed subgroup Δ X ⊆ Π CK may be characterized by a rather simple explicit algorithm. Since the decomposition groups of Π CK at the nonzero cusps — i.e., the cusps whose inertia groups are contained in Δ X [cf. the discussion at the beginning of §1] — are also group-theoretic [cf., e.g., [AbsTopI], Lemma 4.5], the above observation follows immediately from the easily verified fact that the image of any of these decomposition groups associated to nonzero cusps coincides with the image of Π XK in Π CK /Δ X . (ii) In light of the observation of (i), it makes sense to adopt the following convention: Instead of applying the group-theoretic reconstruction algorithm of [AbsTopIII], Theorem 1.9 [cf. also the discussion of [AbsTopIII], Remark 2.8.3], directly to Π CK [or topological groups isomorphic to Π CK ], we shall apply this reconstruction algorithm to the open subgroup Π XK ⊆ Π CK to reconstruct the function field of X K , equipped with its natural Gal(X K /C K ) ∼ = Π CK /Π XK -action. In this context, we shall refer to this approach of applying [AbsTopIII], Theorem 1.9, as the Θ-approach to [AbsTopIII], Theorem 1.9. Note that, for v ∈ V good ⋂ V non (respectively, v ∈ V bad ), one may also adopt a “Θ-approach” to applying [AbsTopIII], Theorem 1.9, to Π Cv or [by applying Corollary 1.2] Π X−→v ,Π C−→v (respectively, to Π tp C or [by applying [EtTh], Proposition 2.4] Π tp v X ). In the present series v of papers, we shall always think of [AbsTopIII], Theorem 1.9 [as well as the other results of [AbsTopIII] that arise as consequences of [AbsTopIII], Theorem 1.9] as being applied to [isomorphs of] Π CK or, for v ∈ V good ⋂ V non (respectively, v ∈ V bad ), Π Cv ,Π X−→v ,Π C−→v (respectively, Π tp C ,Π tp v X ) via the “Θ-approach”. v (iii) Recall from the discussion at the beginning of [EtTh], §2, the tautological extension 1 → Δ Θ → Δ Θ X → Δ ell X → 1 def —whereΔ Θ = [Δ X , Δ X ]/[Δ X , [Δ X , Δ X ]]; Δ Θ def X = Δ X /[Δ X , [Δ X , Δ X ]]; Δ ell def X = Δ ab X . The extension class ∈ H2 (Δ ell X , Δ Θ) of this extension determines a tautological isomorphism M X ∼ → ΔΘ — where we recall from [AbsTopIII], Theorem 1.9, (b), that the module “M X ”of [AbsTopIII], Theorem 1.9, (b) [cf. also [AbsTopIII], Proposition 1.4, (ii)], may be naturally identified with Hom(H 2 (Δ ell X , Ẑ), Ẑ). In particular, we obtain a tautological isomorphism ∼ M X → (l · ΔΘ ) [i.e., since [Δ X :Δ X ]=l]. From the point of view of the theory of the present series of papers, the significance of the “Θ-approach” lies precisely in the existence of this ∼ tautological isomorphism M X → (l · ΔΘ ), which will be applied in [IUTchII] at v ∈
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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